Ray diagram rules
Point a torch at a curved mirror and the reflected beam does something you cannot easily calculate in your head — yet you can draw exactly where it goes. That is the trick behind ray diagrams: instead of algebra, you follow two simple rules for how light bounces off a spherical mirror, and wherever those two reflected rays cross (or seem to cross), that is where the image sits. No formula, no computation — just a ruler, a pencil, and three dependable rules about rays that travel parallel to the axis, through the focus, or through the centre of curvature. This article sets out those rules precisely, shows them at work on a concave and a convex mirror, and flags the mistakes that cost the most marks when sketching one.
Any two of three simple ray rules — parallel-to-axis reflects through F, through-F reflects parallel-to-axis, through-C reflects straight back — locate a mirror's image exactly where the reflected rays actually cross (real image) or appear to cross when extended backward (virtual image).
The core idea
A spherical mirror bends every ray that strikes it, but three particular rays are easy to predict because they interact with the mirror's special points — the pole P, focus F, and centre of curvature C — in a fixed, memorable way.
Rule 1 — parallel to the axis. A ray travelling parallel to the principal axis reflects through F on a concave mirror; on a convex mirror it diverges outward but, extended backward with a dotted line, appears to come from F behind the mirror.
Rule 2 — through or towards the focus. A ray that passes through F on its way to a concave mirror reflects out parallel to the axis. On a convex mirror, a ray aimed towards the virtual F behind the mirror also reflects out parallel to the axis. Rules 1 and 2 are reverse partners of each other.
Rule 3 — through or towards the centre of curvature. A ray through C meets the mirror along the normal (the radius), so it reflects straight back along its own path, retracing itself. This holds for both mirror types, with C sitting behind the mirror for a convex one. You only ever need two of these three rules from the same point on the object (usually its tip). Draw both reflected rays: if they genuinely cross in front of the mirror, that crossing point is a real image (catchable on a screen); if they spread apart and only appear to meet when extended backward with dotted lines, that is a virtual image (seen by looking into the mirror, never caught on a screen).
Why it works
These rules are not arbitrary — each one follows from the plain law of reflection (angle of incidence equals angle of reflection) applied to the geometry of a sphere. A ray parallel to the axis strikes the mirror at some point, and the normal at that point is the radius drawn from C. Because the mirror's curvature is gentle near the axis, the geometry works out so that the reflected ray always crosses the axis at the same point, roughly halfway between the pole and C — that fixed crossing point is, by definition, the focus F. Rule 2 is simply this path run backward: light obeys the same law travelling either direction along a ray, so a ray sent through F retraces the parallel-to-axis path in reverse, leaving parallel to the axis. Rule 3 follows directly from the definition of the normal. At any point on a sphere, the line from that point to the centre C is exactly the normal to the surface there. A ray aimed at C therefore strikes the mirror exactly along the normal, meaning its angle of incidence is zero — and a ray hitting a mirror straight along the normal reflects straight back the way it came, with no bend at all. Because all three rules are just the one law of reflection applied at different points, any two of them must locate the true image consistently — that is why the method needs no formula and no calculation: the geometry alone pins down where the reflected rays converge (or appear to).
Worked example
An object is placed well beyond the centre of curvature of a concave mirror, on the principal axis. Sketch and describe the ray diagram to find its image.
Step 1. Draw the concave mirror with P, C, and F marked on the principal axis, and the object as an upright arrow standing beyond C, as in the figure above.
Step 2. From the tip of the object, apply Rule 1: draw a ray parallel to the axis; after reflection it passes through F.
Step 3. From the same tip, apply Rule 3: draw a ray heading straight for C; it reflects straight back along the same line.
Step 4. The two reflected rays actually cross at a point between C and F, on the same side of the axis as the object. Mark this crossing as the image tip, then drop a perpendicular to the axis to complete the image arrow. Because the reflected rays genuinely cross, the image is real, inverted, and diminished, forming between C and F — the same situation as a concave mirror forming a small, upside-down image of a distant object on a screen.
Competitive stretch
Now place the object very close to a convex mirror instead. Rule 1 from the tip gives a ray that diverges outward but, extended backward with a dotted line, appears to come from F behind the mirror. Rule 3 gives a ray aimed towards C (also behind the mirror) that reflects straight back along the same path. The two solid reflected rays diverge and never meet in front of the mirror, so both must be extended backward until they cross behind it — that crossing point is the image tip. The image is virtual, upright, and diminished, consistent with the general rule that a convex mirror always forms this kind of image, wherever the object is placed.
Common traps
- Forgetting that for a convex mirror (or a concave mirror with the object closer than F), the reflected rays must be extended backward, behind the mirror, to find the virtual image — many students simply stop drawing at the mirror's surface.
- Mixing up Rule 1 and Rule 2: a ray parallel to the axis reflects through the focus, while a ray through the focus reflects parallel to the axis. They are reverse partners, not the same rule stated twice.
- Drawing two random rays from the object and expecting them to give a clean, predictable crossing — only the three named rays (parallel to axis, through focus, through centre of curvature) behave predictably enough to sketch by hand.
- Assuming a solid crossing point always exists. If the reflected rays are diverging — as with any convex mirror, or a concave mirror with the object inside F — there is no real crossing, and the image must be located by extending rays backward with dotted lines.
Ask the tutor: Want to practise sketching a ray diagram for an object placed between the focus and the pole of a concave mirror, or double-check which image type it gives?
Why do we only need two rays instead of drawing many?
Because the mirror is a smooth curved surface, every ray leaving one point on the object and striking anywhere on the mirror reflects so as to pass through (or appear to come from) the same image point. Any two of the three predictable rays are enough to pin that point down exactly — a third ray, if you drew one, would pass through the very same spot, so it adds no new information. Two is the minimum needed to fix a point in a diagram, and these two are simply the easiest to draw accurately.
How do I know if my crossing point is a real image or a virtual one?
Look at whether the reflected rays actually meet as solid lines in front of the mirror, or whether you had to extend them backward with dotted lines behind the mirror to make them meet. A genuine solid-line crossing in front is a real image — light really does converge there, so a screen placed at that spot would catch it. A dotted-line crossing behind the mirror is a virtual image — the rays only look like they came from that point; no light actually passes through it, so no screen there would show anything.
Do I always get the same image type — real or virtual — for a given mirror?
No, it depends on both the mirror and where the object sits. A convex mirror always gives a virtual, upright, diminished image no matter where you place the object, because its reflected rays always diverge. A concave mirror is more varied: an object beyond C gives a real, inverted, diminished image (as in the worked example); an object between C and F also gives a real, inverted, enlarged image; but an object placed inside F gives a virtual, upright, enlarged image, because inside F the reflected rays diverge instead of converging.